Mathematical Model of Armed Criminal Group with Pre-emitive and Repressive Intervention

. Armed Criminal group is one of the problems faced by many countries in the world. Awful behaviour of armed criminal group members can affect a large amount of people. In this paper, we construct a deterministic mathematical model that takes into account persuasive and repressive intervention. We consider crime as a social epidemic. We determine the armed criminal group free equilibrium point and the armed criminal group persistence equilibrium point together with their existence condition. The next generation matrix is used to obtain the basic reproduction number. The local stability conditions of equilibrium points are proved using linearization. We show that the armed criminal group free equilibrium point is globally asymptotically stable under certain condition. Numerical simulations are performed to support our deductive study.


I. INTRODUCTION
Organised crime has a deletrious impact on many countries around the world [1]. They frequently pose serious problems, particularly in urban areas [2]. Armed group, at its most basic level, is an organized group with a clear structure, membership, and the capacity to use aggressivenes in the occupation of its desire [3]. This definition is very general and includes the state armed forces e.g. the police and the army. The armed groups discussed in this article are armed groups that commit crimes. We call it the armed criminal group. Some examples of armed criminal groups are the Yakuza, Triads, drug cartel, and Mafia. They are also called criminal organization [2].
Recently, many mathematical models have been developed to study criminal activity dynamic [4]- [15]. Gonzalez [11] propose a mathematical model of crime by assuming crime as a social epidemic process. The optimal control problem of model of crime is discussed in [6]. A mathematical model of crime that takes into consideration serious and minor criminal activity is discussed in [15]. Jongo [12] study a mathematical model to investigate how minor criminals turn in to major criminals inside and outside of prisons.
Unlike the model previously described, we are specially interested in studying the dynamics of crime caused by individuals or groups (armed criminal groups) when there is persuasive and repressive intervention. Pre-emitive intervention is carried out through education or rehabilitation. On the other hand, repressive intervention is accomplished through punishments e.g. imprisonment.

Model Formulation
We assume that the human population is divided into four disjoint subpopulations. , , , i g P C C Q represent susceptible human, criminals who commit a crime individually, criminals who commit a crime in groups, and humans who choose not to be a criminal, respectively. We assume that recruitment rate of human ( )  is constant.
Susceptible humans ( ) P can become criminals due to interactions with criminals ( ) g C . The effective contact rate between P and g C is denoted  . P reduces because of natural death at rate  . Susceptible humans who have received education about criminal act can supress their desire to commit a crime. The proportion of susceptible humans who receive education and the effectivity of education implementation are denoted  and  , respectively. Based on these assumptions, we get The criminals who commit a crime individually increase because of the new criminals i.e. susceptible humans who are influenced by criminals. After committing several crimes individually, the criminals will decide whether they continue or stop doing crimes. We assume that someone who prefer doing crime will commit a crime in group. On the other hand, someone who prefer to stop doing a crime because of education or other factors go to Q compartment. Sometimes, a criminals are killed due to their crimes. Therefore, we consider using crime induced death rate x  .
Based on these assumptions, we obtain The criminals who commit a crime in group increase because of criminals who usually commit a crime individually join the group. Similar to O compartment, we assume that criminals who commit crimes in group can be killed due to their crimes and can stop doing a crime at rate  . Hence, we get  is the proportion of O who go to M compartment.

Basic Properties
Theorem 1. Solutions of system (-) with non-negative initial value are always non-negative.

Regard as
which leads to a contradiction. Thus, ( ) P t is always non-negative for 0 t  . By similar method, we can show that (

Therorem 2. Solutions of system (-) are bounded
Proof. Let N is the total number of human. Hence, From the system (-), we get Hence, a standard comparison argument provides lim sup ( ) .
This indicates that the solutions of the system are bounded for 0 t  . Based on Theorem 1 and Theorem 2, we obtain feasible region of the system as follows.

Equilibrium points and the basic reproduction number
The armed criminal group free equilibrium point is The armed criminal group persistence equilibrium point is , . x The basic reproduction number is determined by using next generation matrix. From , and 0  , we get

Stability of the armed criminal group free equilibrium points
Proof. We will prove this theorem by using linearization method. The Jacobian matrix of the system (-) is Proof. We will prove this theorem by using Lyapunov direct method. Define : We will prove this theorem by using linearization approach. Subtituting 1  into , we obtain It is convenient to see that

Numerical simulations
We now present numerical simulations to support our theoretical results. We performed two scheme numerical simulations to investigate the impact of pre-emitive and repressive intervention. To study the impact of pre-emitive, we conduct numerical simulation with varying  . To investigate the impact of repressive intervention, we perform numerical simulation with varying x  . We choose the following set parameter values for illustrative purpose only. These parameter values are selected to show the impact of interventions and to verify our theoretical results.

Dynamics of armed criminal group without interventions
In this part, we performed numerical simulation using parameter value that are mentioned in Table 1   is clear that the criminals who commit crimes individually or in group always persist in high density all the time eventhough some criminals quit and never commit a crime again. This condition is certainly worrisome.

Impact of repressive intervention only
In this part, we performed numerical experiment with

Impact of pre-emitive and repressive intervention
In this part, we performed numerical experiment with varying